Problem: Simplify; express your answer in exponential form. Assume $y\neq 0, r\neq 0$. $\dfrac{{(y^{-2}r^{-3})^{-5}}}{{(y^{4}r^{-2})^{3}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-2}r^{-3})^{-5} = (y^{-2})^{-5}(r^{-3})^{-5}}$ On the left, we have ${y^{-2}}$ to the exponent ${-5}$ . Now ${-2 \times -5 = 10}$ , so ${(y^{-2})^{-5} = y^{10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-2}r^{-3})^{-5}}}{{(y^{4}r^{-2})^{3}}} = \dfrac{{y^{10}r^{15}}}{{y^{12}r^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{10}r^{15}}}{{y^{12}r^{-6}}} = \dfrac{{y^{10}}}{{y^{12}}} \cdot \dfrac{{r^{15}}}{{r^{-6}}} = y^{{10} - {12}} \cdot r^{{15} - {(-6)}} = y^{-2}r^{21}$